### TALKS AT MSRI

#### Berkeley, California

I gave the following two talks at the *Stochastic Analysis Seminar
*at the *Mathematical Sciences Research Institute, Berkeley,
California. *The two talks describe recent joint work with Michael Scheutzow.
For a preprint, please see Recent Articles and
Preprints Online.
To download a .dvi file of the two lectures, click on the following:

The Lectures

*FIRST TALK: *
*THE STABLE MANIFOLD THEOREM FOR SDE'S , Part I *

*Wednesday, December 3, 1997, 11:00-12:00 am, MSRI Lecture Hall. *

In this talk, we formulate a local stable manifold theorem for stochastic
differential equations in Euclidean space, driven by multi-dimensional
Brownian motion. We introduce the concept of hyperbolicity for stationary
trajectories of a SDE. This is done using the Oseledec muliplicative ergodic
theorem on the linearized SDE around the stationary solution. Using methods
of (non-linear ergodic theory), we construct a stationary family of stable
and unstable manifolds in a stationary neighborhood around the hyperbolic
stationary trajectory of the non-linear SDE. The stable/unstable manifolds
are dynamically characterized using anticipating stochastic calculus.

*SECOND TALK : *
*THE STABLE MANIFOLD THEOREM FOR SDE'S, Part II *

*Friday, December 5, 1997, 11:00-12:00 am, MSRI Lecture Hall.*

This is a continuation of the talk given on Wednesday, December 3, 1997. We outline the basic ideas underlying the proof of the Stable Manifold Theorem for
SDE's. We discuss the linearization of the SDE along a hyperbolic
stationary solution. The stable and unstable manifolds are constructed
using ideas and techniques from multiplicative ergodic theory that were
developed by David Ruelle in the late seventies. In particular, we
develop estimates of the stochastic flow in a neighborhood of the hyperbolic
stationary solution. Finally, we discuss generalizations to
semimartingale noise, related open problems, and conjectures.

*
List of Members at MSRI (97-98) *

*
Salah's Home Page *

*April 24, 1998.*